Let $\Delta = \{ q \in \mathbb{C}^* | |q| \lt 1\}$, $q \in \Delta$, and $f : \mathbb{C}^*/q^\mathbb{Z} \to E'$ be an isogeny of degree $n$.
Then there exists the unique pair $(a, q')$, where $a$ is a divisor of $n$ and $q'$ satisfies $q^a = q'^b$ for $b = n/a$, such that:
There exists $\mathbb{C}^* / q'^\mathbb{Z} \cong E'$ such that $f$ is the composition $\mathbb{C}^* / q^\mathbb{Z} \to \mathbb{C}^* / q^{a\mathbb{Z}} = \mathbb{C}^* / q'^{b\mathbb{Z}} \to \mathbb{C}^* / q'^\mathbb{Z} \cong E'$. (Where $\mathbb{C}^* / q^\mathbb{Z} \to \mathbb{C}^* / q^{a\mathbb{Z}}$ is $u \mapsto u^a$ and $\mathbb{C}^* / q'^{b\mathbb{Z}} \to \mathbb{C}^* / q'^\mathbb{Z}$ is $u \mapsto u$.)
Here is what I have tried:
Let $f : \mathbb{C}/\Lambda_\tau \to E'$ be an isogeny of degree $n$.
Then this map is isomorphisc to $\mathbb{C}/\Lambda_\tau \to \mathbb{C}/K$, where $K$ is a subgroup of $\mathbb{C}$, containing $\Lambda_\tau$ as a subgroup of index $= n$.
Thus this is a lattice.
So write $K = \omega_1 \mathbb{Z} + \omega_2 \mathbb{Z}$.
Since $K/ \Lambda\tau$ is of index $n$, we may assume $a \omega_1, b \omega_2 \in \Lambda_\tau, n = ab.$
And for such $a,b$, we have $\Lambda_\tau = a \omega_1 \mathbb{Z} + b \omega_2 \mathbb{Z}$.
If $\omega_1 = 1/a, \omega_2 = \tau/b$,
then
$$\mathbb{C}/\Lambda_\tau \xrightarrow{\text{multiplicate by }a} \mathbb{C}/\Lambda_{a\tau}
\xrightarrow{\operatorname{mod}} \mathbb{C}/\Lambda_{\frac{a\tau}{b}}
\xrightarrow[\cong]{\text{multiplicate by }1/a} \mathbb{C}/K $$
is equal to $\mathbb{C}/\Lambda_\tau \to \mathbb{C}/K$, so is isomorphic to $f$, hence ok.
For general $\omega_i$, since the diagram
$\require{AMScd}$
\begin{CD}
\mathbb{C}/\Lambda_\tau @>>> \mathbb{C}/K \\
@V{\times 1/a\omega_1}V{\cong}V @V{\times 1/a\omega_1}V{\cong}V\\
\mathbb{C}/\Lambda_{\frac{b}{a} \frac{\omega_2}{\omega_1}} @>{\times a}>> \mathbb{C}/\Lambda_{\frac{\omega_2}{\omega_1}}
\end{CD}
commutes, the highlighted statement is true if we allow to "compositing isomorphisms with its source and target".
And if $\Im \tau, \Im (\frac{b}{a} \frac{\omega_2}{\omega_1}) \gt 1$, then $q = q'$, where $q = \exp(2 \pi i \tau)$ and $q' = \exp(2 \pi i \frac{b}{a} \frac{\omega_2}{\omega_1})$.
So ok.
But if $|q| \lt e^{-2\pi}$, then $|q'| \ge e^{-2 \pi}$.
So, how can I show the general case?
(I want to show the highlighted statement to show $$\{ \text{ isogenies from } \mathbb{C}^*/q^\mathbb{Z} \text{ of degree }n \} / \sim \\ = \bigcup_{ a | n } \{ q' \in \Delta | q^a = q' ^{n/a} \}, $$ where $E \to E' \sim E \to E'' \iff $ there exists $E' \cong E''$ such that $E \to E''$ is $E \to E' \cong E''$.)